| L(s) = 1 | − 3-s + 3.56·5-s + 7-s + 9-s + 11-s + 0.438·13-s − 3.56·15-s + 2·17-s + 1.56·19-s − 21-s + 7.68·25-s − 27-s + 0.438·29-s + 3.12·31-s − 33-s + 3.56·35-s + 9.80·37-s − 0.438·39-s − 1.12·41-s − 10.2·43-s + 3.56·45-s − 8.68·47-s + 49-s − 2·51-s + 2.87·53-s + 3.56·55-s − 1.56·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.59·5-s + 0.377·7-s + 0.333·9-s + 0.301·11-s + 0.121·13-s − 0.919·15-s + 0.485·17-s + 0.358·19-s − 0.218·21-s + 1.53·25-s − 0.192·27-s + 0.0814·29-s + 0.560·31-s − 0.174·33-s + 0.602·35-s + 1.61·37-s − 0.0702·39-s − 0.175·41-s − 1.56·43-s + 0.530·45-s − 1.26·47-s + 0.142·49-s − 0.280·51-s + 0.395·53-s + 0.480·55-s − 0.206·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.184210221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.184210221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.56T + 5T^{2} \) |
| 13 | \( 1 - 0.438T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 0.438T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 8.68T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 + 4.68T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 1.56T + 67T^{2} \) |
| 71 | \( 1 - 6.24T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 + 6.24T + 79T^{2} \) |
| 83 | \( 1 - 7.12T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.518589989847936184268152410446, −8.549375854566871081684669441821, −7.62741994722189564411576355492, −6.55597401311922205622931187793, −6.08903391970779655645997043621, −5.27716616218414770588336266537, −4.60091343536064424272029901453, −3.22135372980770518582582174202, −2.04204062564621713530009498999, −1.13595700592632520635733462762,
1.13595700592632520635733462762, 2.04204062564621713530009498999, 3.22135372980770518582582174202, 4.60091343536064424272029901453, 5.27716616218414770588336266537, 6.08903391970779655645997043621, 6.55597401311922205622931187793, 7.62741994722189564411576355492, 8.549375854566871081684669441821, 9.518589989847936184268152410446
